We can assume that \(\alpha \le 1/128\)  satisfy eq:alphaepsilon
since otherwise \(\varepsilon \)  is a constant and we can use the list
decodable codes from [1]}. The use
of eq:alphaepsilon will be clear shortly. It becomes a
necessity from round III onward. For rounds I and II, the parameter
\(\alpha \)  will be a constant, but it will be useful to establish the
analysis in more generality now so that subsequent rounds can reuse
it.
